## Final Activity Report Summary - MINCONDINMT (Minimality conditions in model theory)

Wencel developed the model theory of Boolean algebras and their expansions, under a restriction on definable sets, and investigated elimination of imaginaries in Boolean algebras. He also greatly extended the current theory of weakly o-minimal totally ordered first order structures.

Key results concern the good behaviour of topological dimension, and understanding of certain well-behaved weakly o-minimal structures (those with the 'strong cell decomposition property'). He proved that algebraic objects living definably in weakly o-minimal structures carry definably a topology related to their algebraic structure.

These results form a significant contribution to model theory, a branch of mathematical logic, and its connections to topology. Their context is a programme to understand topological first order structures in which definable sets (solution sets of first order formulas) have a natural form, at least in one variable. A striking and unexpected development was a connection to number theory, for example to the Gelfond-Schneider Theorem that if a,b are algebraic numbers with a not equal to 0 or 1 and b not a rational, then a^b, the bth power of a, is trancendental.

Key results concern the good behaviour of topological dimension, and understanding of certain well-behaved weakly o-minimal structures (those with the 'strong cell decomposition property'). He proved that algebraic objects living definably in weakly o-minimal structures carry definably a topology related to their algebraic structure.

These results form a significant contribution to model theory, a branch of mathematical logic, and its connections to topology. Their context is a programme to understand topological first order structures in which definable sets (solution sets of first order formulas) have a natural form, at least in one variable. A striking and unexpected development was a connection to number theory, for example to the Gelfond-Schneider Theorem that if a,b are algebraic numbers with a not equal to 0 or 1 and b not a rational, then a^b, the bth power of a, is trancendental.